Physical Hydrodynamics , Etienne Guyon , Jean-Pierre Hulin , Luc Petit and Catalin D. Mitescu Oxford U. Press, New York, 2001. $100.00, $50.00 paper (505 pp.). ISBN 0-19-851746-7, ISBN 0-19-851745-9 paper
Turbulence is one of the most difficult problems in classical physics. In spite of tremendous progress on both the experimental and theoretical fronts, it still is far from being completely understood. The apocryphal discussion between Wolfgang Pauli and God, in which God considers turbulence a harder problem than calculating the value of the fine structure constant, is still valid in the 21st century. Some of the leading physicists and mathematicians of the 20th century contributed to the study of turbulence: Richard Feynman, Lev Landau, Werner Heisenberg, Andrei Kolmogorov, Jean Leray, Eberhard Hopf, David Ruelle, Robert Kraichnan, among many others.
The two books reviewed here are written at quite different levels. Navier–Stokes Equations and Turbulence , by Ciprian Foias, Oscar Manley, Ricardo Rosa, and Roger Temam, is an exposition of the known mathematical facts about turbulence and is aimed primarily at physicists and engineers; Physical Hydrodynamics , by Etienne Guyon, Jean-Pierre Hulin, Luc Petit, and Catalin D. Mitescu, is a highly original introductory text. Nonetheless, the two books do complement each other nicely.
I will devote most of this review to the first book. But I want to bring the second one to the attention of the physics community because of its interesting and unusual approach to the subject, emphasized in a beautiful foreword by Pierre-Gilles de Gennes.
Navier–Stokes Equations and Turbulence is part of the series Encyclopedia of Mathematics and its Applications; its explicit aim is to serve as an introduction for physicists and engineers to recent developments in the mathematics of turbulence. It is also intended to be an introduction for mathematicians to some of the physics and engineering aspects of turbulence. From a theoretical physicist’s point of view, one of the book’s most attractive features is the fact that the authors tried to interpret seemingly artificial mathematical assumptions in physical terms: for example, a particular choice of norms in Sobolev spaces is described as setting bounds on the energy or the enstrophy (the square of vorticity).
The book is written in such a way that the details of the proofs can be easily skipped without losing an understanding of the main reasoning. For example, the fact that solutions of the Navier–Stokes equations in two and three dimensions become more regular than their initial conditions leads to the need to consider so-called Gevrey spaces, and this is clearly explained. One highly original feature of the book is its insistence on statistical aspects of turbulence without getting lost in hairy details. The details are relegated to the more technical appendices of each of the chapters.
Here is a brief overview of the book’s contents: Chapter 1 sets the background, with a brief account of the fundamental contributions of Kolmogorov and Kraichnan. Chapter 2 discusses the mathematical theory of the Navier–Stokes equations and the physical interpretation of the various spaces used. Chapter 3 discusses the finite dimensionality of turbulent flows—the fact that, in a harmonic decomposition of the flow, the high-wavenumber components decay so rapidly that the energy is carried by a finite number of components with low wavenumbers. The corresponding Kolmogorov and Kraichnan “guesstimates” lead to finite but very large numbers for these dimensions; as a consequence, low-dimensional truncations, such as Edward Lorenz’s three-dimensional model, are only a caricature of the real chaotic character of the flow. The second half of the chapter deals with the fractal dimensions of the “attractors”—the long-term evolution of the flows.
Chapter 4 deals in detail with stationary statistical solutions of the Navier–Stokes equations, time averages, and attractors, focusing especially on the problem of “equality” of time and ensemble averages without appealing to ergodic hypotheses. It is the hardest and most interesting of the chapters: It deals with nonstationary statistical solutions of the Navier–Stokes equations and their relation to the conventional statistical approaches. As far as I know, this chapter is the first account of a mathematical validation, based on the Navier–Stokes equations, of the results obtained by conventional, heuristic approaches to turbulence. This includes a justification of the Kolmogorov spectrum and of intermittency, but is far from solving all fluid dynamics problems.
Physical Hydrodynamics is a rather original introduction to fluid dynamics and emphasizes the molecular and microscopic origins of fluid phenomena. It is well illustrated with pictures resulting from actual experiments and numerical simulations, which are cleverly used to motivate the more mathematical developments. The authors have been involved in experimental and numerical modeling of fluid flows and related subjects, and many illustrations in the book are from their original work.
The book starts with discussions of the solid–liquid transition, plastic flows, and macroscopic transport coefficients. It exploits models and numerical simulation to develop the reader’s intuition. The text describes in detail experimental aspects, as well as such “unorthodox” fluids as Bingham fluids, thixotropic fluids, bubbles, smoke rings, and others. Another nice feature is the authors’ exploitation of electromagnetic analogies, particularly in explaining vorticity and the motion of vortex filaments. Hydrodynamic instabilities and transition to turbulence are beautifully illustrated. In short, this is a book that can be read with pleasure by college seniors, graduate students, or professors familiar with the subject. It nicely complements Navier–Stokes Equations and Turbulence , which could have benefited from an index of notations and a more detailed author index. Although, like the authors, I am not a native English speaker, I noticed some Franco–Romanian syntactic structures and nonstandard punctuation in both books. The copy-editors of both could have done more to make the text-flow more laminar.
